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real number

1 Definition

A Cauchy sequence of rational numbers is a sequence{xi},i=0,1,2,…formulae-sequencesubscriptxii012normal-…\{x_{i}\},\ i=0,1,2,\dots of rational numbers with the property that, for every rational number ϵ>0ϵ0\epsilon>0, there exists a natural numberNNN such that, for all natural numbers n,m>NnmNn,m>N, the absolute value|xn-xm|subscriptxnsubscriptxm|x_{n}-x_{m}|satisfies|xn-xm|<ϵsubscriptxnsubscriptxmϵ|x_{n}-x_{m}|<\epsilon.

There is an ordering relation on ℝℝ\mathbb{R}, defined by {xi}≤{yi}subscriptxisubscriptyi\{x_{i}\}\leq\{y_{i}\} if either {xi}∼{yi}similar-tosubscriptxisubscriptyi\{x_{i}\}\sim\{y_{i}\} or there exists a natural number NNN such that xn<ynsubscriptxnsubscriptynx_{n}<y_{n} for all n>NnNn>N. This definition is well-defined and does not depend on the choice of Cauchy sequences used to represent the equivalence classes.

Equivalence classes of decimal sequences (sequences consisting of natural numbers between 0 and 9, and a single decimal point), where two decimal sequences are equivalent if they are identical, or if one has an infinite tail of 9’s, the other has an infinite tail of 0’s, and the leading portion of the first sequence is one lower than the leading portion of the second.

2.

Dedekind cuts of rational numbers (that is, subsets SSS of ℚℚ\mathbb{Q} with the property that, if a∈SaSa\in S and b<abab<a, then b∈SbSb\in S).

3.

The real numbers can also be defined as the unique (up to isomorphism) ordered field satisfying the least upper bound property, after one has proved that such a field exists and is unique up to isomorphism.

A sequence (xn)subscriptxn(x_{n}) of real numbers is called a Cauchy sequence if for any
ε>0ε0\varepsilon>0
there exists an integerNNN (possibly depending on εε\varepsilon) such that the
distance|xn-xm|subscriptxnsubscriptxm|x_{n}-x_{m}| is less than εε\varepsilon provided that nnn and mmm are
both greater than NNN. In other
words, a sequence is a Cauchy sequence if its elements xnsubscriptxnx_{n}eventually come
and remain arbitrarily close to each other.

A sequence (xn)subscriptxn(x_{n})converges to the limit xxx if for any ε>0ε0\varepsilon>0
there exists an integer NNN (possibly depending on εε\varepsilon) such that the
distance |xn-x|subscriptxnx|x_{n}-x| is less than εε\varepsilon provided that nnn is greater
than NNN. In other words, a sequence has limit xxx if its elements eventually
come and remain arbitrarily close to xxx.

Note that the rationals are not complete. For example, the sequence 111, 1.41.41.4,
1.411.411.41, 1.4141.4141.414, 1.41421.41421.4142, 1.414211.414211.41421, …normal-…\ldots is Cauchy but it does not
converge to a rational number. (In the real numbers, in contrast, it converges
to the square root of 222.)

The existence of limits of Cauchy sequences is what makes calculus work and is
of great practical use. The standard numerical test to determine if a sequence
has a limit is to test if it is a Cauchy sequence, as the limit is typically
not known in advance.

can be made arbitrarily small by choosing NNN sufficiently large. This proves
that the sequence is Cauchy, so we know that the sequence converges even if we
don’t know ahead of time what the limit is.

3 “The complete ordered field”

The real numbers are often described as “the complete ordered field,” a phrase
that can be interpreted in several ways.

First, an order can be lattice complete. It’s easy to see that no ordered field
can be lattice complete, because it can have no largest element (given any
element zzz, z+1z1z+1 is larger), so this is not the sense that is meant.

Additionally, an order can be Dedekind-complete, as defined in the Definitions section. The uniqueness result at the end of that section justifies using the
word “the” in the phrase “complete ordered field” when this is the sense of
“complete” that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction
starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.

These two notions of completeness ignore the field structure. However, an
ordered group (and a field is a group under the operations of addition and
subtraction) defines a uniform structure, and uniform structures have a notion
of completeness (topology); the description in the Completeness section above
is a special case. (We refer to the notion of completeness in uniform spaces
rather than the related and better known notion for metric spaces, since the
definition of metric space relies on already having a characterisation of the
real numbers.) It is not true that ℝℝ\mathbb{R} is the only uniformly complete
ordered
field, but it is the only uniformly complete Archimedean field, and indeed one
often hears the phrase “complete Archimedean field” instead of “complete
ordered field.” Since it can be proved that any uniformly complete Archimedean
field must also be Dedekind complete (and vice versa, of course), this
justifies using “the” in the phrase “the complete Archimedean field.” This
sense of completeness is most closely related to the construction of the reals
from Cauchy sequences (the construction carried out in full in this article),
since it starts with an Archimedean field (the rationals) and forms the uniform
completion of it in a standard way.

But the original use of the phrase “complete Archimedean field” was by David
Hilbert, who meant still something else by it. He meant that the real numbers
form the largest Archimedean field in the sense that every other Archimedean
field is a subfield of ℝℝ\mathbb{R}. Thus ℝℝ\mathbb{R} is “complete” in the
sense that nothing
further can be added to it without making it no longer an Archimedean field.
This sense of completeness is most closely related to the construction of the
reals from surreal numbers, since that construction starts with a proper class
that contains every ordered field (the surreals) and then selects from it the
largest Archimedean subfield.

Mathematics Subject Classification

Comments

I've always understood the real numbers as having a relatively simple axiomatic definition (e.g. [0]). I'll admit I didn't properly understand this entry, but it didn't even seem to acknowledge the existance of this kind of definition. Is it worth me writing such a definition, either for inclusion on this article, or as a seperate article?

> Further, 'sum' and 'product' are closed operations in
> $\mathbb{R}$. The reference given by Adam does not make
> mention to that fact.

actually it does - the first couple of lines say:

For every pair of real numbers a, b \in R there is a unique real number a + b, called their â€˜sumâ€™.
For every pair of real numbers a, b \in R there is a unique real number a Â· b, called their â€˜productâ€™.